Differential Forms and Boundary Integral Equations for Maxwell-Type Problems

TL;DR

This paper develops boundary-integral equations for Maxwell-type problems using differential-form calculus, providing a general framework applicable to various dimensions and boundary regularities, and offering new insights into their properties.

Contribution

It introduces a Sobolev-space framework and integral representations for Maxwell-type problems in a differential-form setting, broadening the mathematical tools available for these equations.

Findings

Established a general Sobolev-space framework for Maxwell-type problems.

Derived integral transformations and fundamental solutions.

Provided new insights into properties and symmetries of boundary-integral equations.

Abstract

We present boundary-integral equations for Maxwell-type problems in a differential-form setting. Maxwell-type problems are governed by the differential equation $(\delta\mathrm{d}-k^2)\omega = 0$, where $k\in\mathbb{C}$ holds, subject to some restrictions. This problem class generalizes $\textbf{curl}\,\textbf{curl}$- and $\mathrm{div}\,\textbf{grad}$-types of problems in three dimensions. The goal of the paper is threefold: 1) Establish the Sobolev-space framework in the full generality of differential-form calculus on a smooth manifold of arbitrary dimension and with Lipschitz boundary. 2) Introduce integral transformations and fundamental solutions, and derive a representation formula for Maxwell-type problems. 3) Leverage the power of differential-form calculus to gain insight into properties and inherent symmetries of boundary-integral equations of Maxwell-type.